How To Really Solve This Tricky Algebra Problem (VII)

elcome to the seventh entry in the tricky algebra problem series. Unlike the last couple of entries, we are dealing with a more advanced problem this time around. I hope you find it as engaging as I did! Without any further ado, let us begin.

You are presented with five unknowns: a, b, c, d, and e, all of which can only take positive integer values. Next, you are given the following equation:

(a*b*c*d*e) = (a + b + c + d + e)

Given these conditions, your challenge is to compute maximum possible value of max{a, b, c, d, e}. In plain words, your solution will need to be the maximum possible integer value of all five variables involved (a single value).

Do you think you can solve this?

Spoiler Alert:

Beyond this section, I will be explicitly discussing the solution to this problem. So, if you wish to solve this problem on your own, I suggest that you pause reading this essay at this point and go ahead with your solution.

Once you are done with your attempt, you may come back and continue reading to compare our approaches.

This essay is supported by Generatebg

---

Problem Setup and Starting Point

The key to start solving this problem, in my opinion, is to notice the implicit relationship between the five variables. Since all five are unknown, we can be certain that at least one of them will take the maximum integer value. Consequently, we may suppose the following condition:

a ≤ b ≤ c ≤ d ≤ e

As soon as we suppose this condition, another subtle realisation emerges. Since we are only interested in the maximum value of all the integers, we now need only to compute the maximum value of e, that is, max{e}.

With this at the back of our minds, I’d like to show you two approaches to solving this problem:

1. A slower but more general approach.

2. A faster but more specific approach.

---

A Generalised Approach to Solving the Tricky Algebra Problem

Let us begin with the condition that we had supposed:

a ≤ b ≤ c ≤ d ≤ e

Next, let us turn the problem on its head and consider the smallest possible integer value (which is 1) we can get away with for each variable. If we consider (e = 1), then by our supposition, a = b = c = d = 1.

Under these conditions, we arrive at the following issue:

(a*b*c*d*e) = 1

(a + b + c + d + e) = 5

(a*b*c*d*e) cannot be equal to (a + b + c + d + e)

So, we know for sure that e cannot be equal to 1. But what about d? If we consider (d = 1), then e is our only unknown in the problem. Consequently, we arrive at the following equation system:

(a*b*c*d*e) = (1*1*1*1*e) = e

(a + b + c + d + e) = (1 + 1 + 1 + 1 + e) = 4 + e

As we can clearly see, (4 + e) cannot be equal to e.

Working Out Two Unknowns

Let us keep going: consider (c = 1). Then, by our supposition, a = b = c = 1. As a result, we arrive at the following expressions:

Now this is something we can work with. When we plug these results into the original equation, we get the following result:

If you look at the left-hand side of the final expression we have, you might notice that we can rearrange the terms and express them as the following multiplication operation:

This equation directly leads us to our solution. We can clearly see that the value of d cannot be 1. If (d = 1), then the left-hand side of the equation goes to zero. So, our first realisation is that d must be at least 2 (that is, d ≥ 1).

If we consider the minimum value of d (d = 2), then the only other acceptable value of e is 5 (that is, e = 5). As a result, we arrive at the following range of possibilities and in turn, our final answer:

We also establish that the following is one of our solutions:

{a, b, c, d, e} = {1, 1, 1, 2, 5}

But note that this problem has several solutions. This fact becomes clearer when you look at the alternative approach that I am about to show you next.

---

A Specific Approach to Solving the Tricky Algebra Problem

Tostart, let us make the same supposition that we had made the first time around:

a ≤ b ≤ c ≤ d ≤ e

So, we need to only find the maximum value for e. Also, based on this inequality, we can be certain of the following inequality:

e < (a + b + c + d + e)

We now have a lower bound in terms of e. Can we also have an upper bound in terms of e? The answer is: yes.

Based on our supposition, the highest possible value for each variable is ‘e’. As a result, we arrive at the following inequality:

e < (a + b + c + d + e) ≤ (5*e)

Let us now plug the given equation into the above inequality. As a result, we would get the following inequality:

e < (a*b*c*d*e) ≤ (5*e)

When we divide this inequality by e throughout, we get the following result:

1 < (a*b*c*d) ≤ 5

Considering our original supposition, this inequality leads to the following possibilities for the values of a, b, c, and d:

{a, b, c, d} = {1, 1, 1, 2}, {1, 1, 1, 3}, {1, 1, 1, 4}, {1, 1, 1, 5}, or {1, 1, 2, 2}

This consequently leads us to max{e} = 5.

Note that not all of the {a, b, c, d} combinations above lead to valid solutions for the given equation. We still need to enforce the given equation to arrive at solutions. Examples of solutions are: {1, 1, 1, 2, 5} and {1, 1, 2, 2, 2}.

There we go. This approach was quicker, but the first approach we took is generalisable to any number of unknowns!

---

Source: Andreescu T., and Feng Z.

If you’d like to get notified when interesting content gets published here, consider subscribing.

Further reading that might interest you: How To Really Solve This Packing Spheres Puzzle? and The Story Of The Rockstar Mathematician Who Never Lived.

If you would like to support me as an author, consider contributing on Patreon.